Comments on Theorem 3.2 With the primal problem in (6) in the paper, Theorem 3.2 provides Additionally, [27] presents the optimization w.r.t. a single projection direction in Therefore, our KSVD is more general in the data setups. Remark 3.3, we show that the values can be regarded as playing the role of the dual variables Using data-dependent projection weights does not affect the derivation of the shifted eigenvalue problem in the dual. With the derivations of the primal-dual optimization problems above, the primal-dual model representation of our KSVD problem can be provided correspondingly. Lemma 4.2 evaluates the objective value Moreover, as in the proof of Theorem 3.2, we note that the regularization coefficient This section provides the implementation details of all experiments included in the paper. This will be illustrated in details in the following.Algorithm 1 Learning with Primal-AttentionRequire: X:= [ x UEA Time Series The UEA time series benchmark [31] consists of 30 datasets. Following the setup in [11], we select 10 datasets for evaluation.

Recently, a new line of works has emerged to understand and improve self-attention in Transformers by treating it as a kernel machine. However, existing works apply the methods for symmetric kernels to the asymmetric self-attention, resulting in a nontrivial gap between the analytical understanding and numerical implementation. In this paper, we provide a new perspective to represent and optimize self-attention through asymmetric Kernel Singular Value Decomposition (KSVD), which is also motivated by the low-rank property of self-attention normally observed in deep layers. Through asymmetric KSVD, i) a primal-dual representation of self-attention is formulated, where the optimization objective is cast to maximize the projection variances in the attention outputs; ii) a novel attention mechanism, i.e., Primal-Attention, is proposed via the primal representation of KSVD, avoiding explicit computation of the kernel matrix in the dual; iii) with KKT conditions, we prove that the stationary solution to the KSVD optimization in Primal-Attention yields a zero-value objective. In this manner, KSVD optimization can be implemented by simply minimizing a regularization loss, so that low-rank property is promoted without extra decomposition. Numerical experiments show state-of-the-art performance of our Primal-Attention with improved efficiency. Moreover, we demonstrate that the deployed KSVD optimization regularizes Primal-Attention with a sharper singular value decay than that of the canonical self-attention, further verifying the great potential of our method. To the best of our knowledge, this is the first work that provides a primal-dual representation for the asymmetric kernel in self-attention and successfully applies it to modelling and optimization.

Comments on Theorem 3.2 With the primal problem in (6) in the paper, Theorem 3.2 provides Additionally, [27] presents the optimization w.r.t. a single projection direction in Therefore, our KSVD is more general in the data setups. Remark 3.3, we show that the values can be regarded as playing the role of the dual variables Using data-dependent projection weights does not affect the derivation of the shifted eigenvalue problem in the dual. With the derivations of the primal-dual optimization problems above, the primal-dual model representation of our KSVD problem can be provided correspondingly. Lemma 4.2 evaluates the objective value Moreover, as in the proof of Theorem 3.2, we note that the regularization coefficient This section provides the implementation details of all experiments included in the paper. This will be illustrated in details in the following.Algorithm 1 Learning with Primal-AttentionRequire: X:= [ x UEA Time Series The UEA time series benchmark [31] consists of 30 datasets. Following the setup in [11], we select 10 datasets for evaluation.

Recently, a new line of works has emerged to understand and improve self-attention in Transformers by treating it as a kernel machine. However, existing works apply the methods for symmetric kernels to the asymmetric self-attention, resulting in a nontrivial gap between the analytical understanding and numerical implementation. In this paper, we provide a new perspective to represent and optimize self-attention through asymmetric Kernel Singular Value Decomposition (KSVD), which is also motivated by the low-rank property of self-attention normally observed in deep layers. Through asymmetric KSVD, i) a primal-dual representation of self-attention is formulated, where the optimization objective is cast to maximize the projection variances in the attention outputs; ii) a novel attention mechanism, i.e., Primal-Attention, is proposed via the primal representation of KSVD, avoiding explicit computation of the kernel matrix in the dual; iii) with KKT conditions, we prove that the stationary solution to the KSVD optimization in Primal-Attention yields a zero-value objective. In this manner, KSVD optimization can be implemented by simply minimizing a regularization loss, so that low-rank property is promoted without extra decomposition.

Recently, a new line of works has emerged to understand and improve self-attention in Transformers by treating it as a kernel machine. However, existing works apply the methods for symmetric kernels to the asymmetric self-attention, resulting in a nontrivial gap between the analytical understanding and numerical implementation. In this paper, we provide a new perspective to represent and optimize self-attention through asymmetric Kernel Singular Value Decomposition (KSVD), which is also motivated by the low-rank property of self-attention normally observed in deep layers. Through asymmetric KSVD, $i$) a primal-dual representation of self-attention is formulated, where the optimization objective is cast to maximize the projection variances in the attention outputs; $ii$) a novel attention mechanism, i.e., Primal-Attention, is proposed via the primal representation of KSVD, avoiding explicit computation of the kernel matrix in the dual; $iii$) with KKT conditions, we prove that the stationary solution to the KSVD optimization in Primal-Attention yields a zero-value objective. In this manner, KSVD optimization can be implemented by simply minimizing a regularization loss, so that low-rank property is promoted without extra decomposition. Numerical experiments show state-of-the-art performance of our Primal-Attention with improved efficiency. Moreover, we demonstrate that the deployed KSVD optimization regularizes Primal-Attention with a sharper singular value decay than that of the canonical self-attention, further verifying the great potential of our method. To the best of our knowledge, this is the first work that provides a primal-dual representation for the asymmetric kernel in self-attention and successfully applies it to modeling and optimization.